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We introduce the notion of compatible actions in the context of splitextensions of finite dimensional Lie algebras over a field. Using compatibleactions, we construct a new resolution to compute the cohomology of semi-directproducts of Lie algebras. We also give an alternative way to construct theHochschild-Serre spectral sequence associated to a split extension of finitedimensional Lie algebras and obtain a sharper bound for the length of thisspectral sequence.

On cohomology of split Lie algebra extensions